Limit theorems for a class of critical superprocesses with stable branching

We consider a critical superprocess {X;Pμ} with general spatial motion and spatially dependent stable branching mechanism with lowest stable index γ0>1. We first show that, under some conditions, Pμ(|Xt|≠0) converges to 0 as t→∞ and is regularly varying with index (γ0−1)−1. Then we show that, for...

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Bibliographic Details
Published in:Stochastic processes and their applications 2020-07, Vol.130 (7), p.4358-4391
Main Authors: Ren, Yan-Xia, Song, Renming, Sun, Zhenyao
Format: Article
Language:English
Subjects:
Critical superprocess
Intrinsic ultracontractivity
Regular variation
Scaling limit
Stable branching
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Summary:We consider a critical superprocess {X;Pμ} with general spatial motion and spatially dependent stable branching mechanism with lowest stable index γ0>1. We first show that, under some conditions, Pμ(|Xt|≠0) converges to 0 as t→∞ and is regularly varying with index (γ0−1)−1. Then we show that, for a large class of non-negative testing functions f, the distribution of {Xt(f);Pμ(⋅|‖Xt‖≠0)}, after appropriate rescaling, converges weakly to a positive random variable z(γ0−1) with Laplace transform E[e−uz(γ0−1)]=1−(1+u−(γ0−1))−1∕(γ0−1).
ISSN:0304-4149
1879-209X
DOI:10.1016/j.spa.2020.01.001